Implementation of Point Source

作者: jjx323 | 来源:发表于2020-12-02 15:01 被阅读0次

Junxiong Jia, 2020.12.02

Introduction

For solving inverse problems, we often meet point source when tacking adjoint equations, e.g., the following equation
$-\Delta u - e^{2q}u = -\frac{1}{\sigma^2}\sum_{j=1}^{K}(w(x)-d_j)\delta(x-x_j) \quad\text{ in }\Omega,$
where $\{x_j\}_{j=1}^{K}$ are some measurement points, $w(x)$ is usually some function obtained from forward equation. In the following, we explain that how to implement the point source term on the right-hand side of the above equation.

Measurement Matrix

Usually, for a function in $L^2(\Omega)$ ( $\Omega$ is some open bounded domain), it can be decomposed by some fixed orthonormal basis $\{ \varphi_{i} \}_{i=1}^{\infty}$ . In the finite element method, we project functions on a finite number of basis as follow
$u(x) \approx \sum_{i=1}^{N}u_i \varphi_i(x).$
And once we specify the coefficients $\{u_i\}_{i=1}^{N}$ , the function $u(x)$ will be specified. We introduce the following measurement matrix
$S = \left[\begin{matrix} \varphi_1(x_1) & \cdots & \varphi_N(x_1) \\ \vdots & & \vdots \\ \varphi_1(x_K) & \cdots & \varphi_N(x_K) \end{matrix}\right].$
The matrix has the following property
$\left[\begin{matrix} u(x_1) \\ \vdots \\ u(x_K) \end{matrix}\right] = S \left[\begin{matrix} u_1 \\ \vdots \\ u_N \end{matrix}\right].$

Implement of the Point Source

Now we can give specific suggestions for implementing the point source. For discretization, we usually introduce the following quantities
$f_k := \int_{\Omega}\sum_{j=1}^{K}(w(x)-d_j)\delta(x-x_j)\varphi_{k}(x)dx$
with $k = 1,2,\cdots, N$ . Actually, we can reduce $f_k$ by some simple calculations as follow:
$f_k = \sum_{j=1}^{K}(w(x_j)-d_j)\varphi_k(x_j).$
With these calculations, we find that
$\left[\begin{matrix} f_1 \\ \vdots \\ f_N \end{matrix}\right] = \left[\begin{matrix} \varphi_1(x_1) & \cdots & \varphi_1(x_K) \\ \vdots & & \vdots \\ \varphi_{N}(x_1) & \cdots & \varphi_{N}(x_K) \end{matrix}\right] \left[\begin{matrix} w(x_1)-d_1 \\ \vdots \\ w(x_K)-d_K \end{matrix}\right] = S^T \left[\begin{matrix} w(x_1)-d_1 \\ \vdots \\ w(x_K)-d_K \end{matrix}\right].$
Denote $d = (d_1,\cdots,d_K)^T$ , $f=(f_1,\cdots,f_N)^T$ and $w=(w_1,\cdots,w_N)^T$ (coefficients of the function $w$ ), we can reduce the above formula as follows:
$f = S^T(Sw - d),$
which is convenient for implementations by some numerical algorithm, e.g., finite element method.

Implementation of Point Source

Introduction

Measurement Matrix

Implement of the Point Source

相关文章

网友评论

延伸阅读

深度阅读

栏目导航

热点阅读